Properties

Label 8001.2.a.t
Level 8001
Weight 2
Character orbit 8001.a
Self dual Yes
Analytic conductor 63.888
Analytic rank 0
Dimension 16
CM No
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 8001.a (trivial)

Newform invariants

Self dual: Yes
Analytic conductor: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{15}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( 1 - \beta_{11} ) q^{5} \) \(- q^{7}\) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{8} \) \(+O(q^{10})\) \( q\) \( + \beta_{1} q^{2} \) \( + ( 1 + \beta_{2} ) q^{4} \) \( + ( 1 - \beta_{11} ) q^{5} \) \(- q^{7}\) \( + ( -1 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{5} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{8} \) \( + ( -1 + 2 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{5} - \beta_{10} - \beta_{11} + \beta_{12} + \beta_{15} ) q^{10} \) \( + ( 2 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{6} + \beta_{9} - \beta_{11} - \beta_{13} ) q^{11} \) \( + ( \beta_{2} - \beta_{3} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - \beta_{13} ) q^{13} \) \( -\beta_{1} q^{14} \) \( + ( -1 - \beta_{2} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{11} + 2 \beta_{14} + \beta_{15} ) q^{16} \) \( + ( -\beta_{1} + \beta_{3} + \beta_{4} - \beta_{7} + \beta_{8} - \beta_{9} + 2 \beta_{10} + 2 \beta_{11} + \beta_{14} ) q^{17} \) \( + ( -2 - \beta_{2} + \beta_{8} - \beta_{11} + 2 \beta_{14} ) q^{19} \) \( + ( 2 - \beta_{1} - \beta_{3} + \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} + \beta_{14} ) q^{20} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} + \beta_{10} + \beta_{12} + \beta_{13} + \beta_{14} + \beta_{15} ) q^{22} \) \( + ( 3 - \beta_{1} + 2 \beta_{2} - 3 \beta_{3} - \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + \beta_{9} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{14} - \beta_{15} ) q^{23} \) \( + ( 1 - \beta_{1} - \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} - \beta_{13} - 2 \beta_{15} ) q^{25} \) \( + ( 1 + \beta_{1} - \beta_{3} - \beta_{5} - \beta_{6} + \beta_{9} - \beta_{10} - 2 \beta_{11} + \beta_{14} + \beta_{15} ) q^{26} \) \( + ( -1 - \beta_{2} ) q^{28} \) \( + ( -1 - \beta_{2} + \beta_{5} - \beta_{6} + \beta_{8} + 2 \beta_{14} ) q^{29} \) \( + ( -1 - \beta_{1} - 4 \beta_{3} - 2 \beta_{4} + \beta_{5} - \beta_{7} + \beta_{9} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} - \beta_{15} ) q^{31} \) \( + ( -1 - \beta_{2} + 4 \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{10} + \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + \beta_{15} ) q^{32} \) \( + ( -2 + 2 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - \beta_{5} + 2 \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} ) q^{34} \) \( + ( -1 + \beta_{11} ) q^{35} \) \( + ( \beta_{1} + \beta_{3} + \beta_{6} + 2 \beta_{7} + \beta_{8} - \beta_{9} - \beta_{10} + \beta_{13} ) q^{37} \) \( + ( 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - 2 \beta_{8} + \beta_{12} + \beta_{13} + \beta_{15} ) q^{38} \) \( + ( -3 + 3 \beta_{1} - 2 \beta_{2} + \beta_{3} + 2 \beta_{4} - \beta_{5} + \beta_{6} - \beta_{8} - \beta_{10} - \beta_{11} + 2 \beta_{12} + 2 \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{40} \) \( + ( 3 - \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} + \beta_{5} + 2 \beta_{6} - \beta_{7} + \beta_{8} + \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{13} - \beta_{15} ) q^{41} \) \( + ( 2 + \beta_{2} - \beta_{3} - \beta_{4} - \beta_{6} - \beta_{8} - \beta_{9} - 2 \beta_{11} - \beta_{12} - \beta_{13} - \beta_{14} ) q^{43} \) \( + ( 1 + \beta_{1} + 3 \beta_{3} + \beta_{4} - \beta_{5} - \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{12} + 2 \beta_{13} + 2 \beta_{14} + 2 \beta_{15} ) q^{44} \) \( + ( 4 + 4 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} - \beta_{6} + \beta_{7} - \beta_{8} + \beta_{9} - \beta_{10} - \beta_{11} - 3 \beta_{12} - 3 \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{46} \) \( + ( 2 - \beta_{2} + 2 \beta_{3} + \beta_{4} + \beta_{5} + 2 \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} - \beta_{15} ) q^{47} \) \(+ q^{49}\) \( + ( -1 + 4 \beta_{1} - \beta_{2} - 2 \beta_{4} + \beta_{6} - \beta_{9} - 4 \beta_{10} - \beta_{11} + 2 \beta_{12} - 2 \beta_{14} + 2 \beta_{15} ) q^{50} \) \( + ( 1 + 4 \beta_{1} - \beta_{2} + 2 \beta_{3} + \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{8} - \beta_{9} - \beta_{11} + \beta_{12} + 2 \beta_{13} + \beta_{14} + 3 \beta_{15} ) q^{52} \) \( + ( 2 - 2 \beta_{1} + 3 \beta_{2} - 2 \beta_{3} - \beta_{5} - \beta_{6} - 2 \beta_{8} + 2 \beta_{10} + \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{53} \) \( + ( 2 + \beta_{1} + 3 \beta_{2} - \beta_{3} - 2 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} + \beta_{8} - \beta_{11} - \beta_{12} - \beta_{14} + \beta_{15} ) q^{55} \) \( + ( 1 - \beta_{1} + \beta_{2} - \beta_{3} - \beta_{4} + \beta_{5} - \beta_{12} - \beta_{13} - \beta_{14} - \beta_{15} ) q^{56} \) \( + ( 1 + 2 \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - 2 \beta_{8} + \beta_{10} + 2 \beta_{11} + \beta_{12} + 2 \beta_{13} ) q^{58} \) \( + ( 5 + 2 \beta_{1} + 3 \beta_{2} - \beta_{4} - \beta_{5} + \beta_{7} - \beta_{8} - \beta_{10} + \beta_{11} - \beta_{12} - 2 \beta_{13} - 3 \beta_{14} - 2 \beta_{15} ) q^{59} \) \( + ( 4 - \beta_{1} + 6 \beta_{2} - 4 \beta_{3} - 4 \beta_{4} + \beta_{5} + \beta_{6} + \beta_{7} - 2 \beta_{8} + 3 \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - \beta_{12} - 3 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{61} \) \( + ( -2 \beta_{1} + \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} + \beta_{7} - 2 \beta_{8} - 2 \beta_{9} - \beta_{10} + 2 \beta_{11} - 2 \beta_{14} ) q^{62} \) \( + ( -3 - 2 \beta_{2} + 6 \beta_{3} + 2 \beta_{4} - \beta_{5} + 3 \beta_{8} - \beta_{9} + 3 \beta_{10} + 3 \beta_{11} + 3 \beta_{12} + 3 \beta_{13} + 2 \beta_{14} ) q^{64} \) \( + ( -3 + 4 \beta_{1} - \beta_{2} + \beta_{3} - \beta_{6} + \beta_{7} - \beta_{9} - 3 \beta_{10} - \beta_{11} + 2 \beta_{12} + 3 \beta_{13} + 3 \beta_{15} ) q^{65} \) \( + ( 2 + 2 \beta_{3} + 2 \beta_{4} - \beta_{5} - \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} - \beta_{14} - \beta_{15} ) q^{67} \) \( + ( 4 - 2 \beta_{1} + 3 \beta_{2} + 3 \beta_{3} + \beta_{5} - \beta_{6} + \beta_{7} + \beta_{8} + \beta_{9} + \beta_{10} + 2 \beta_{11} + 2 \beta_{13} + 2 \beta_{14} ) q^{68} \) \( + ( 1 - 2 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{5} + \beta_{10} + \beta_{11} - \beta_{12} - \beta_{15} ) q^{70} \) \( + ( 6 - \beta_{1} + 2 \beta_{2} - 4 \beta_{3} - 3 \beta_{4} + \beta_{5} - \beta_{6} - 2 \beta_{7} - \beta_{8} + 2 \beta_{9} - \beta_{10} - 2 \beta_{11} - \beta_{12} - 2 \beta_{13} ) q^{71} \) \( + ( -1 - \beta_{1} - \beta_{2} + \beta_{5} - \beta_{6} - 2 \beta_{9} + 3 \beta_{10} + \beta_{11} - \beta_{13} ) q^{73} \) \( + ( 4 + 3 \beta_{2} - 5 \beta_{3} - 3 \beta_{4} + \beta_{5} + 2 \beta_{6} - 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - 3 \beta_{11} - 2 \beta_{12} - 3 \beta_{13} - 4 \beta_{14} - \beta_{15} ) q^{74} \) \( + ( \beta_{3} + \beta_{4} - \beta_{5} - 2 \beta_{6} - \beta_{7} + 2 \beta_{8} + \beta_{9} + 2 \beta_{10} + \beta_{11} - \beta_{12} + \beta_{14} ) q^{76} \) \( + ( -2 + \beta_{1} - \beta_{2} + \beta_{3} + \beta_{4} - \beta_{6} - \beta_{9} + \beta_{11} + \beta_{13} ) q^{77} \) \( + ( -6 - 4 \beta_{2} + 3 \beta_{3} + 4 \beta_{4} - 2 \beta_{5} - \beta_{6} - \beta_{7} - \beta_{10} + 3 \beta_{12} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{79} \) \( + ( \beta_{1} + 2 \beta_{2} - 2 \beta_{4} + \beta_{5} - \beta_{6} + 3 \beta_{8} - \beta_{10} + \beta_{11} + \beta_{13} + \beta_{14} + 2 \beta_{15} ) q^{80} \) \( + ( -4 + 4 \beta_{1} - 3 \beta_{2} + 3 \beta_{3} + \beta_{4} + 2 \beta_{6} + \beta_{8} - 2 \beta_{9} - 3 \beta_{10} + 3 \beta_{12} + \beta_{13} - \beta_{14} + 2 \beta_{15} ) q^{82} \) \( + ( 6 + 4 \beta_{2} + 2 \beta_{3} - \beta_{5} - \beta_{6} + 3 \beta_{7} - \beta_{8} - \beta_{9} + \beta_{10} + 2 \beta_{11} + \beta_{13} - 3 \beta_{14} - \beta_{15} ) q^{83} \) \( + ( -2 \beta_{1} + \beta_{2} + \beta_{3} - \beta_{4} - \beta_{6} + \beta_{8} + 2 \beta_{10} - \beta_{12} + \beta_{13} + \beta_{14} ) q^{85} \) \( + ( 3 \beta_{1} - \beta_{2} - 2 \beta_{3} - \beta_{4} - 2 \beta_{5} - \beta_{6} + \beta_{7} - \beta_{8} - 2 \beta_{10} - 3 \beta_{11} - \beta_{12} - 2 \beta_{13} - 2 \beta_{14} - \beta_{15} ) q^{86} \) \( + ( -4 + 6 \beta_{1} - 6 \beta_{2} + 6 \beta_{3} + 3 \beta_{4} - 3 \beta_{5} - \beta_{6} - \beta_{7} + \beta_{8} - \beta_{10} + 6 \beta_{12} + 5 \beta_{13} + 5 \beta_{14} + 4 \beta_{15} ) q^{88} \) \( + ( 4 + 3 \beta_{1} + \beta_{2} - 5 \beta_{3} - 2 \beta_{4} - 3 \beta_{8} + \beta_{9} - 2 \beta_{10} - 2 \beta_{11} - 2 \beta_{12} - \beta_{13} - 2 \beta_{14} + \beta_{15} ) q^{89} \) \( + ( -\beta_{2} + \beta_{3} + \beta_{8} - \beta_{9} + \beta_{10} + \beta_{11} + \beta_{13} ) q^{91} \) \( + ( 1 + 3 \beta_{1} + \beta_{2} - \beta_{3} + \beta_{4} - 2 \beta_{5} - 2 \beta_{6} + \beta_{7} - 4 \beta_{8} - 2 \beta_{9} - \beta_{10} + \beta_{11} - 2 \beta_{12} - 4 \beta_{14} ) q^{92} \) \( + ( 3 + \beta_{1} - 2 \beta_{4} + \beta_{5} + \beta_{6} - 2 \beta_{8} + \beta_{12} - 3 \beta_{14} - \beta_{15} ) q^{94} \) \( + ( -\beta_{4} + 2 \beta_{5} - \beta_{7} + 3 \beta_{8} - \beta_{9} + 3 \beta_{11} + \beta_{13} + \beta_{14} - \beta_{15} ) q^{95} \) \( + ( 2 + 3 \beta_{2} + \beta_{5} + 2 \beta_{7} - 3 \beta_{8} - \beta_{11} - \beta_{13} - \beta_{14} ) q^{97} \) \( + \beta_{1} q^{98} \) \(+O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \(16q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 16q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut +\mathstrut O(q^{10}) \) \(16q \) \(\mathstrut +\mathstrut 2q^{2} \) \(\mathstrut +\mathstrut 12q^{4} \) \(\mathstrut +\mathstrut 9q^{5} \) \(\mathstrut -\mathstrut 16q^{7} \) \(\mathstrut +\mathstrut 6q^{8} \) \(\mathstrut -\mathstrut 2q^{10} \) \(\mathstrut +\mathstrut 22q^{11} \) \(\mathstrut -\mathstrut 4q^{13} \) \(\mathstrut -\mathstrut 2q^{14} \) \(\mathstrut +\mathstrut 12q^{16} \) \(\mathstrut +\mathstrut 18q^{17} \) \(\mathstrut -\mathstrut 15q^{19} \) \(\mathstrut +\mathstrut 40q^{20} \) \(\mathstrut -\mathstrut 11q^{22} \) \(\mathstrut +\mathstrut 5q^{23} \) \(\mathstrut +\mathstrut 15q^{25} \) \(\mathstrut +\mathstrut 24q^{26} \) \(\mathstrut -\mathstrut 12q^{28} \) \(\mathstrut +\mathstrut 12q^{29} \) \(\mathstrut -\mathstrut 32q^{31} \) \(\mathstrut +\mathstrut 9q^{32} \) \(\mathstrut -\mathstrut 14q^{34} \) \(\mathstrut -\mathstrut 9q^{35} \) \(\mathstrut -\mathstrut 2q^{37} \) \(\mathstrut -\mathstrut 3q^{38} \) \(\mathstrut -\mathstrut 14q^{40} \) \(\mathstrut +\mathstrut 45q^{41} \) \(\mathstrut -\mathstrut 3q^{43} \) \(\mathstrut +\mathstrut 54q^{44} \) \(\mathstrut +\mathstrut 49q^{47} \) \(\mathstrut +\mathstrut 16q^{49} \) \(\mathstrut +\mathstrut 6q^{50} \) \(\mathstrut +\mathstrut 38q^{52} \) \(\mathstrut -\mathstrut 16q^{53} \) \(\mathstrut +\mathstrut 7q^{55} \) \(\mathstrut -\mathstrut 6q^{56} \) \(\mathstrut +\mathstrut 16q^{58} \) \(\mathstrut +\mathstrut 35q^{59} \) \(\mathstrut -\mathstrut 11q^{61} \) \(\mathstrut -\mathstrut 17q^{62} \) \(\mathstrut -\mathstrut 2q^{64} \) \(\mathstrut -\mathstrut 14q^{65} \) \(\mathstrut +\mathstrut 17q^{67} \) \(\mathstrut +\mathstrut 71q^{68} \) \(\mathstrut +\mathstrut 2q^{70} \) \(\mathstrut +\mathstrut 81q^{71} \) \(\mathstrut -\mathstrut 15q^{73} \) \(\mathstrut -\mathstrut 13q^{74} \) \(\mathstrut +\mathstrut 14q^{76} \) \(\mathstrut -\mathstrut 22q^{77} \) \(\mathstrut -\mathstrut 34q^{79} \) \(\mathstrut +\mathstrut 33q^{80} \) \(\mathstrut -\mathstrut 14q^{82} \) \(\mathstrut +\mathstrut 39q^{83} \) \(\mathstrut -\mathstrut 17q^{85} \) \(\mathstrut -\mathstrut 36q^{86} \) \(\mathstrut +\mathstrut 61q^{88} \) \(\mathstrut +\mathstrut 32q^{89} \) \(\mathstrut +\mathstrut 4q^{91} \) \(\mathstrut -\mathstrut 37q^{92} \) \(\mathstrut +\mathstrut 13q^{94} \) \(\mathstrut +\mathstrut 33q^{95} \) \(\mathstrut -\mathstrut 4q^{97} \) \(\mathstrut +\mathstrut 2q^{98} \) \(\mathstrut +\mathstrut O(q^{100}) \)

Basis of coefficient ring in terms of a root \(\nu\) of \(x^{16}\mathstrut -\mathstrut \) \(2\) \(x^{15}\mathstrut -\mathstrut \) \(20\) \(x^{14}\mathstrut +\mathstrut \) \(38\) \(x^{13}\mathstrut +\mathstrut \) \(155\) \(x^{12}\mathstrut -\mathstrut \) \(275\) \(x^{11}\mathstrut -\mathstrut \) \(593\) \(x^{10}\mathstrut +\mathstrut \) \(957\) \(x^{9}\mathstrut +\mathstrut \) \(1177\) \(x^{8}\mathstrut -\mathstrut \) \(1655\) \(x^{7}\mathstrut -\mathstrut \) \(1150\) \(x^{6}\mathstrut +\mathstrut \) \(1279\) \(x^{5}\mathstrut +\mathstrut \) \(474\) \(x^{4}\mathstrut -\mathstrut \) \(280\) \(x^{3}\mathstrut -\mathstrut \) \(83\) \(x^{2}\mathstrut +\mathstrut \) \(x\mathstrut +\mathstrut \) \(1\):

\(\beta_{0}\)\(=\)\( 1 \)
\(\beta_{1}\)\(=\)\( \nu \)
\(\beta_{2}\)\(=\)\( \nu^{2} - 3 \)
\(\beta_{3}\)\(=\)\((\)\(-\)\(116093\) \(\nu^{15}\mathstrut +\mathstrut \) \(228893\) \(\nu^{14}\mathstrut +\mathstrut \) \(2347523\) \(\nu^{13}\mathstrut -\mathstrut \) \(4412026\) \(\nu^{12}\mathstrut -\mathstrut \) \(18385016\) \(\nu^{11}\mathstrut +\mathstrut \) \(32533154\) \(\nu^{10}\mathstrut +\mathstrut \) \(70696203\) \(\nu^{9}\mathstrut -\mathstrut \) \(115867538\) \(\nu^{8}\mathstrut -\mathstrut \) \(138531569\) \(\nu^{7}\mathstrut +\mathstrut \) \(205570931\) \(\nu^{6}\mathstrut +\mathstrut \) \(126910371\) \(\nu^{5}\mathstrut -\mathstrut \) \(162346281\) \(\nu^{4}\mathstrut -\mathstrut \) \(41913198\) \(\nu^{3}\mathstrut +\mathstrut \) \(34856907\) \(\nu^{2}\mathstrut +\mathstrut \) \(6759511\) \(\nu\mathstrut -\mathstrut \) \(128712\)\()/239255\)
\(\beta_{4}\)\(=\)\((\)\(118542\) \(\nu^{15}\mathstrut -\mathstrut \) \(282272\) \(\nu^{14}\mathstrut -\mathstrut \) \(2211727\) \(\nu^{13}\mathstrut +\mathstrut \) \(5170999\) \(\nu^{12}\mathstrut +\mathstrut \) \(15614719\) \(\nu^{11}\mathstrut -\mathstrut \) \(35484771\) \(\nu^{10}\mathstrut -\mathstrut \) \(52971862\) \(\nu^{9}\mathstrut +\mathstrut \) \(114412902\) \(\nu^{8}\mathstrut +\mathstrut \) \(91508371\) \(\nu^{7}\mathstrut -\mathstrut \) \(178289744\) \(\nu^{6}\mathstrut -\mathstrut \) \(79215634\) \(\nu^{5}\mathstrut +\mathstrut \) \(121751459\) \(\nu^{4}\mathstrut +\mathstrut \) \(32149637\) \(\nu^{3}\mathstrut -\mathstrut \) \(26089138\) \(\nu^{2}\mathstrut -\mathstrut \) \(4109094\) \(\nu\mathstrut +\mathstrut \) \(299938\)\()/239255\)
\(\beta_{5}\)\(=\)\((\)\(246643\) \(\nu^{15}\mathstrut -\mathstrut \) \(556793\) \(\nu^{14}\mathstrut -\mathstrut \) \(4734663\) \(\nu^{13}\mathstrut +\mathstrut \) \(10433381\) \(\nu^{12}\mathstrut +\mathstrut \) \(34632721\) \(\nu^{11}\mathstrut -\mathstrut \) \(74008394\) \(\nu^{10}\mathstrut -\mathstrut \) \(121920963\) \(\nu^{9}\mathstrut +\mathstrut \) \(250361358\) \(\nu^{8}\mathstrut +\mathstrut \) \(213453449\) \(\nu^{7}\mathstrut -\mathstrut \) \(416685796\) \(\nu^{6}\mathstrut -\mathstrut \) \(167476866\) \(\nu^{5}\mathstrut +\mathstrut \) \(306694906\) \(\nu^{4}\mathstrut +\mathstrut \) \(40444858\) \(\nu^{3}\mathstrut -\mathstrut \) \(64193652\) \(\nu^{2}\mathstrut -\mathstrut \) \(2974546\) \(\nu\mathstrut +\mathstrut \) \(1167877\)\()/239255\)
\(\beta_{6}\)\(=\)\((\)\(-\)\(416133\) \(\nu^{15}\mathstrut +\mathstrut \) \(848713\) \(\nu^{14}\mathstrut +\mathstrut \) \(8205093\) \(\nu^{13}\mathstrut -\mathstrut \) \(16039026\) \(\nu^{12}\mathstrut -\mathstrut \) \(62183706\) \(\nu^{11}\mathstrut +\mathstrut \) \(115239599\) \(\nu^{10}\mathstrut +\mathstrut \) \(229254488\) \(\nu^{9}\mathstrut -\mathstrut \) \(397334578\) \(\nu^{8}\mathstrut -\mathstrut \) \(425687974\) \(\nu^{7}\mathstrut +\mathstrut \) \(679630526\) \(\nu^{6}\mathstrut +\mathstrut \) \(361459511\) \(\nu^{5}\mathstrut -\mathstrut \) \(519191331\) \(\nu^{4}\mathstrut -\mathstrut \) \(101394473\) \(\nu^{3}\mathstrut +\mathstrut \) \(112731547\) \(\nu^{2}\mathstrut +\mathstrut \) \(10770031\) \(\nu\mathstrut -\mathstrut \) \(1601912\)\()/239255\)
\(\beta_{7}\)\(=\)\((\)\(-\)\(420868\) \(\nu^{15}\mathstrut +\mathstrut \) \(831558\) \(\nu^{14}\mathstrut +\mathstrut \) \(8486798\) \(\nu^{13}\mathstrut -\mathstrut \) \(16040641\) \(\nu^{12}\mathstrut -\mathstrut \) \(66156781\) \(\nu^{11}\mathstrut +\mathstrut \) \(118505764\) \(\nu^{10}\mathstrut +\mathstrut \) \(252322388\) \(\nu^{9}\mathstrut -\mathstrut \) \(423867643\) \(\nu^{8}\mathstrut -\mathstrut \) \(486918174\) \(\nu^{7}\mathstrut +\mathstrut \) \(758894461\) \(\nu^{6}\mathstrut +\mathstrut \) \(431479801\) \(\nu^{5}\mathstrut -\mathstrut \) \(611916381\) \(\nu^{4}\mathstrut -\mathstrut \) \(127789533\) \(\nu^{3}\mathstrut +\mathstrut \) \(142341487\) \(\nu^{2}\mathstrut +\mathstrut \) \(15316331\) \(\nu\mathstrut -\mathstrut \) \(2488852\)\()/239255\)
\(\beta_{8}\)\(=\)\((\)\(86516\) \(\nu^{15}\mathstrut -\mathstrut \) \(193256\) \(\nu^{14}\mathstrut -\mathstrut \) \(1686213\) \(\nu^{13}\mathstrut +\mathstrut \) \(3671187\) \(\nu^{12}\mathstrut +\mathstrut \) \(12597346\) \(\nu^{11}\mathstrut -\mathstrut \) \(26550396\) \(\nu^{10}\mathstrut -\mathstrut \) \(45669957\) \(\nu^{9}\mathstrut +\mathstrut \) \(92287029\) \(\nu^{8}\mathstrut +\mathstrut \) \(83323677\) \(\nu^{7}\mathstrut -\mathstrut \) \(159344793\) \(\nu^{6}\mathstrut -\mathstrut \) \(69683347\) \(\nu^{5}\mathstrut +\mathstrut \) \(123008780\) \(\nu^{4}\mathstrut +\mathstrut \) \(19521756\) \(\nu^{3}\mathstrut -\mathstrut \) \(27300111\) \(\nu^{2}\mathstrut -\mathstrut \) \(2391162\) \(\nu\mathstrut +\mathstrut \) \(403609\)\()/47851\)
\(\beta_{9}\)\(=\)\((\)\(-\)\(558146\) \(\nu^{15}\mathstrut +\mathstrut \) \(1242406\) \(\nu^{14}\mathstrut +\mathstrut \) \(10825416\) \(\nu^{13}\mathstrut -\mathstrut \) \(23508057\) \(\nu^{12}\mathstrut -\mathstrut \) \(80263942\) \(\nu^{11}\mathstrut +\mathstrut \) \(169127043\) \(\nu^{10}\mathstrut +\mathstrut \) \(287448331\) \(\nu^{9}\mathstrut -\mathstrut \) \(583976571\) \(\nu^{8}\mathstrut -\mathstrut \) \(513565088\) \(\nu^{7}\mathstrut +\mathstrut \) \(1000551217\) \(\nu^{6}\mathstrut +\mathstrut \) \(412351657\) \(\nu^{5}\mathstrut -\mathstrut \) \(766742622\) \(\nu^{4}\mathstrut -\mathstrut \) \(104612936\) \(\nu^{3}\mathstrut +\mathstrut \) \(169828364\) \(\nu^{2}\mathstrut +\mathstrut \) \(14520502\) \(\nu\mathstrut -\mathstrut \) \(2398759\)\()/239255\)
\(\beta_{10}\)\(=\)\((\)\(609731\) \(\nu^{15}\mathstrut -\mathstrut \) \(1339991\) \(\nu^{14}\mathstrut -\mathstrut \) \(11975331\) \(\nu^{13}\mathstrut +\mathstrut \) \(25605932\) \(\nu^{12}\mathstrut +\mathstrut \) \(90323612\) \(\nu^{11}\mathstrut -\mathstrut \) \(186671853\) \(\nu^{10}\mathstrut -\mathstrut \) \(331094336\) \(\nu^{9}\mathstrut +\mathstrut \) \(655602681\) \(\nu^{8}\mathstrut +\mathstrut \) \(610664783\) \(\nu^{7}\mathstrut -\mathstrut \) \(1145738667\) \(\nu^{6}\mathstrut -\mathstrut \) \(514021097\) \(\nu^{5}\mathstrut +\mathstrut \) \(894448927\) \(\nu^{4}\mathstrut +\mathstrut \) \(142265856\) \(\nu^{3}\mathstrut -\mathstrut \) \(197861309\) \(\nu^{2}\mathstrut -\mathstrut \) \(18219637\) \(\nu\mathstrut +\mathstrut \) \(2939939\)\()/239255\)
\(\beta_{11}\)\(=\)\((\)\(-\)\(614167\) \(\nu^{15}\mathstrut +\mathstrut \) \(1330387\) \(\nu^{14}\mathstrut +\mathstrut \) \(12063962\) \(\nu^{13}\mathstrut -\mathstrut \) \(25378599\) \(\nu^{12}\mathstrut -\mathstrut \) \(90934424\) \(\nu^{11}\mathstrut +\mathstrut \) \(184614846\) \(\nu^{10}\mathstrut +\mathstrut \) \(332488247\) \(\nu^{9}\mathstrut -\mathstrut \) \(646723747\) \(\nu^{8}\mathstrut -\mathstrut \) \(608767076\) \(\nu^{7}\mathstrut +\mathstrut \) \(1127337289\) \(\nu^{6}\mathstrut +\mathstrut \) \(501713704\) \(\nu^{5}\mathstrut -\mathstrut \) \(878849284\) \(\nu^{4}\mathstrut -\mathstrut \) \(127489287\) \(\nu^{3}\mathstrut +\mathstrut \) \(195153183\) \(\nu^{2}\mathstrut +\mathstrut \) \(14029589\) \(\nu\mathstrut -\mathstrut \) \(2942448\)\()/239255\)
\(\beta_{12}\)\(=\)\((\)\(-\)\(673264\) \(\nu^{15}\mathstrut +\mathstrut \) \(1527129\) \(\nu^{14}\mathstrut +\mathstrut \) \(13105079\) \(\nu^{13}\mathstrut -\mathstrut \) \(29129163\) \(\nu^{12}\mathstrut -\mathstrut \) \(97629743\) \(\nu^{11}\mathstrut +\mathstrut \) \(211857962\) \(\nu^{10}\mathstrut +\mathstrut \) \(351724054\) \(\nu^{9}\mathstrut -\mathstrut \) \(742060049\) \(\nu^{8}\mathstrut -\mathstrut \) \(632322762\) \(\nu^{7}\mathstrut +\mathstrut \) \(1294215363\) \(\nu^{6}\mathstrut +\mathstrut \) \(508299188\) \(\nu^{5}\mathstrut -\mathstrut \) \(1013016703\) \(\nu^{4}\mathstrut -\mathstrut \) \(121901624\) \(\nu^{3}\mathstrut +\mathstrut \) \(232294846\) \(\nu^{2}\mathstrut +\mathstrut \) \(12764513\) \(\nu\mathstrut -\mathstrut \) \(5229541\)\()/239255\)
\(\beta_{13}\)\(=\)\((\)\(907437\) \(\nu^{15}\mathstrut -\mathstrut \) \(1921932\) \(\nu^{14}\mathstrut -\mathstrut \) \(18024552\) \(\nu^{13}\mathstrut +\mathstrut \) \(36966909\) \(\nu^{12}\mathstrut +\mathstrut \) \(137780884\) \(\nu^{11}\mathstrut -\mathstrut \) \(271915646\) \(\nu^{10}\mathstrut -\mathstrut \) \(512234712\) \(\nu^{9}\mathstrut +\mathstrut \) \(966370827\) \(\nu^{8}\mathstrut +\mathstrut \) \(954511391\) \(\nu^{7}\mathstrut -\mathstrut \) \(1714237034\) \(\nu^{6}\mathstrut -\mathstrut \) \(797268019\) \(\nu^{5}\mathstrut +\mathstrut \) \(1362704444\) \(\nu^{4}\mathstrut +\mathstrut \) \(198561717\) \(\nu^{3}\mathstrut -\mathstrut \) \(308068303\) \(\nu^{2}\mathstrut -\mathstrut \) \(20676569\) \(\nu\mathstrut +\mathstrut \) \(5318248\)\()/239255\)
\(\beta_{14}\)\(=\)\((\)\(-\)\(183898\) \(\nu^{15}\mathstrut +\mathstrut \) \(398886\) \(\nu^{14}\mathstrut +\mathstrut \) \(3621479\) \(\nu^{13}\mathstrut -\mathstrut \) \(7620756\) \(\nu^{12}\mathstrut -\mathstrut \) \(27413174\) \(\nu^{11}\mathstrut +\mathstrut \) \(55548928\) \(\nu^{10}\mathstrut +\mathstrut \) \(100969347\) \(\nu^{9}\mathstrut -\mathstrut \) \(195111036\) \(\nu^{8}\mathstrut -\mathstrut \) \(187405313\) \(\nu^{7}\mathstrut +\mathstrut \) \(341269962\) \(\nu^{6}\mathstrut +\mathstrut \) \(159092271\) \(\nu^{5}\mathstrut -\mathstrut \) \(267285935\) \(\nu^{4}\mathstrut -\mathstrut \) \(44412449\) \(\nu^{3}\mathstrut +\mathstrut \) \(59983859\) \(\nu^{2}\mathstrut +\mathstrut \) \(5138082\) \(\nu\mathstrut -\mathstrut \) \(1000302\)\()/47851\)
\(\beta_{15}\)\(=\)\((\)\(929511\) \(\nu^{15}\mathstrut -\mathstrut \) \(2103041\) \(\nu^{14}\mathstrut -\mathstrut \) \(18058381\) \(\nu^{13}\mathstrut +\mathstrut \) \(39940442\) \(\nu^{12}\mathstrut +\mathstrut \) \(134317747\) \(\nu^{11}\mathstrut -\mathstrut \) \(288743733\) \(\nu^{10}\mathstrut -\mathstrut \) \(483981381\) \(\nu^{9}\mathstrut +\mathstrut \) \(1003060396\) \(\nu^{8}\mathstrut +\mathstrut \) \(875314583\) \(\nu^{7}\mathstrut -\mathstrut \) \(1730295122\) \(\nu^{6}\mathstrut -\mathstrut \) \(721664127\) \(\nu^{5}\mathstrut +\mathstrut \) \(1334031662\) \(\nu^{4}\mathstrut +\mathstrut \) \(195849826\) \(\nu^{3}\mathstrut -\mathstrut \) \(296868004\) \(\nu^{2}\mathstrut -\mathstrut \) \(24599592\) \(\nu\mathstrut +\mathstrut \) \(5430944\)\()/239255\)
\(1\)\(=\)\(\beta_0\)
\(\nu\)\(=\)\(\beta_{1}\)
\(\nu^{2}\)\(=\)\(\beta_{2}\mathstrut +\mathstrut \) \(3\)
\(\nu^{3}\)\(=\)\(\beta_{15}\mathstrut +\mathstrut \) \(\beta_{14}\mathstrut +\mathstrut \) \(\beta_{13}\mathstrut +\mathstrut \) \(\beta_{12}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(\beta_{4}\mathstrut +\mathstrut \) \(\beta_{3}\mathstrut -\mathstrut \) \(\beta_{2}\mathstrut +\mathstrut \) \(5\) \(\beta_{1}\mathstrut -\mathstrut \) \(1\)
\(\nu^{4}\)\(=\)\(\beta_{15}\mathstrut +\mathstrut \) \(2\) \(\beta_{14}\mathstrut -\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(\beta_{8}\mathstrut -\mathstrut \) \(\beta_{7}\mathstrut +\mathstrut \) \(5\) \(\beta_{2}\mathstrut +\mathstrut \) \(13\)
\(\nu^{5}\)\(=\)\(9\) \(\beta_{15}\mathstrut +\mathstrut \) \(9\) \(\beta_{14}\mathstrut +\mathstrut \) \(10\) \(\beta_{13}\mathstrut +\mathstrut \) \(10\) \(\beta_{12}\mathstrut +\mathstrut \) \(\beta_{11}\mathstrut +\mathstrut \) \(\beta_{10}\mathstrut -\mathstrut \) \(9\) \(\beta_{5}\mathstrut +\mathstrut \) \(10\) \(\beta_{4}\mathstrut +\mathstrut \) \(12\) \(\beta_{3}\mathstrut -\mathstrut \) \(9\) \(\beta_{2}\mathstrut +\mathstrut \) \(28\) \(\beta_{1}\mathstrut -\mathstrut \) \(9\)
\(\nu^{6}\)\(=\)\(10\) \(\beta_{15}\mathstrut +\mathstrut \) \(22\) \(\beta_{14}\mathstrut +\mathstrut \) \(3\) \(\beta_{13}\mathstrut +\mathstrut \) \(3\) \(\beta_{12}\mathstrut -\mathstrut \) \(7\) \(\beta_{11}\mathstrut +\mathstrut \) \(3\) \(\beta_{10}\mathstrut +\mathstrut \) \(9\) \(\beta_{9}\mathstrut +\mathstrut \) \(13\) \(\beta_{8}\mathstrut -\mathstrut \) \(10\) \(\beta_{7}\mathstrut -\mathstrut \) \(\beta_{5}\mathstrut +\mathstrut \) \(2\) \(\beta_{4}\mathstrut +\mathstrut \) \(6\) \(\beta_{3}\mathstrut +\mathstrut \) \(24\) \(\beta_{2}\mathstrut +\mathstrut \) \(63\)
\(\nu^{7}\)\(=\)\(68\) \(\beta_{15}\mathstrut +\mathstrut \) \(72\) \(\beta_{14}\mathstrut +\mathstrut \) \(81\) \(\beta_{13}\mathstrut +\mathstrut \) \(81\) \(\beta_{12}\mathstrut +\mathstrut \) \(15\) \(\beta_{11}\mathstrut +\mathstrut \) \(16\) \(\beta_{10}\mathstrut -\mathstrut \) \(\beta_{9}\mathstrut +\mathstrut \) \(3\) \(\beta_{8}\mathstrut -\mathstrut \) \(3\) \(\beta_{7}\mathstrut +\mathstrut \) \(\beta_{6}\mathstrut -\mathstrut \) \(66\) \(\beta_{5}\mathstrut +\mathstrut \) \(80\) \(\beta_{4}\mathstrut +\mathstrut \) \(107\) \(\beta_{3}\mathstrut -\mathstrut \) \(70\) \(\beta_{2}\mathstrut +\mathstrut \) \(166\) \(\beta_{1}\mathstrut -\mathstrut \) \(71\)
\(\nu^{8}\)\(=\)\(81\) \(\beta_{15}\mathstrut +\mathstrut \) \(193\) \(\beta_{14}\mathstrut +\mathstrut \) \(49\) \(\beta_{13}\mathstrut +\mathstrut \) \(45\) \(\beta_{12}\mathstrut -\mathstrut \) \(30\) \(\beta_{11}\mathstrut +\mathstrut \) \(46\) \(\beta_{10}\mathstrut +\mathstrut \) \(64\) \(\beta_{9}\mathstrut +\mathstrut \) \(124\) \(\beta_{8}\mathstrut -\mathstrut \) \(81\) \(\beta_{7}\mathstrut -\mathstrut \) \(16\) \(\beta_{5}\mathstrut +\mathstrut \) \(35\) \(\beta_{4}\mathstrut +\mathstrut \) \(92\) \(\beta_{3}\mathstrut +\mathstrut \) \(109\) \(\beta_{2}\mathstrut +\mathstrut \) \(\beta_{1}\mathstrut +\mathstrut \) \(320\)
\(\nu^{9}\)\(=\)\(492\) \(\beta_{15}\mathstrut +\mathstrut \) \(555\) \(\beta_{14}\mathstrut +\mathstrut \) \(618\) \(\beta_{13}\mathstrut +\mathstrut \) \(614\) \(\beta_{12}\mathstrut +\mathstrut \) \(161\) \(\beta_{11}\mathstrut +\mathstrut \) \(171\) \(\beta_{10}\mathstrut -\mathstrut \) \(14\) \(\beta_{9}\mathstrut +\mathstrut \) \(49\) \(\beta_{8}\mathstrut -\mathstrut \) \(45\) \(\beta_{7}\mathstrut +\mathstrut \) \(12\) \(\beta_{6}\mathstrut -\mathstrut \) \(460\) \(\beta_{5}\mathstrut +\mathstrut \) \(599\) \(\beta_{4}\mathstrut +\mathstrut \) \(864\) \(\beta_{3}\mathstrut -\mathstrut \) \(517\) \(\beta_{2}\mathstrut +\mathstrut \) \(1021\) \(\beta_{1}\mathstrut -\mathstrut \) \(537\)
\(\nu^{10}\)\(=\)\(619\) \(\beta_{15}\mathstrut +\mathstrut \) \(1562\) \(\beta_{14}\mathstrut +\mathstrut \) \(537\) \(\beta_{13}\mathstrut +\mathstrut \) \(469\) \(\beta_{12}\mathstrut -\mathstrut \) \(45\) \(\beta_{11}\mathstrut +\mathstrut \) \(490\) \(\beta_{10}\mathstrut +\mathstrut \) \(428\) \(\beta_{9}\mathstrut +\mathstrut \) \(1041\) \(\beta_{8}\mathstrut -\mathstrut \) \(614\) \(\beta_{7}\mathstrut -\mathstrut \) \(2\) \(\beta_{6}\mathstrut -\mathstrut \) \(172\) \(\beta_{5}\mathstrut +\mathstrut \) \(398\) \(\beta_{4}\mathstrut +\mathstrut \) \(969\) \(\beta_{3}\mathstrut +\mathstrut \) \(448\) \(\beta_{2}\mathstrut +\mathstrut \) \(22\) \(\beta_{1}\mathstrut +\mathstrut \) \(1658\)
\(\nu^{11}\)\(=\)\(3511\) \(\beta_{15}\mathstrut +\mathstrut \) \(4202\) \(\beta_{14}\mathstrut +\mathstrut \) \(4602\) \(\beta_{13}\mathstrut +\mathstrut \) \(4524\) \(\beta_{12}\mathstrut +\mathstrut \) \(1495\) \(\beta_{11}\mathstrut +\mathstrut \) \(1567\) \(\beta_{10}\mathstrut -\mathstrut \) \(134\) \(\beta_{9}\mathstrut +\mathstrut \) \(549\) \(\beta_{8}\mathstrut -\mathstrut \) \(469\) \(\beta_{7}\mathstrut +\mathstrut \) \(96\) \(\beta_{6}\mathstrut -\mathstrut \) \(3169\) \(\beta_{5}\mathstrut +\mathstrut \) \(4376\) \(\beta_{4}\mathstrut +\mathstrut \) \(6668\) \(\beta_{3}\mathstrut -\mathstrut \) \(3728\) \(\beta_{2}\mathstrut +\mathstrut \) \(6447\) \(\beta_{1}\mathstrut -\mathstrut \) \(3961\)
\(\nu^{12}\)\(=\)\(4639\) \(\beta_{15}\mathstrut +\mathstrut \) \(12147\) \(\beta_{14}\mathstrut +\mathstrut \) \(5003\) \(\beta_{13}\mathstrut +\mathstrut \) \(4231\) \(\beta_{12}\mathstrut +\mathstrut \) \(822\) \(\beta_{11}\mathstrut +\mathstrut \) \(4497\) \(\beta_{10}\mathstrut +\mathstrut \) \(2816\) \(\beta_{9}\mathstrut +\mathstrut \) \(8185\) \(\beta_{8}\mathstrut -\mathstrut \) \(4524\) \(\beta_{7}\mathstrut -\mathstrut \) \(46\) \(\beta_{6}\mathstrut -\mathstrut \) \(1592\) \(\beta_{5}\mathstrut +\mathstrut \) \(3796\) \(\beta_{4}\mathstrut +\mathstrut \) \(8793\) \(\beta_{3}\mathstrut +\mathstrut \) \(1437\) \(\beta_{2}\mathstrut +\mathstrut \) \(322\) \(\beta_{1}\mathstrut +\mathstrut \) \(8626\)
\(\nu^{13}\)\(=\)\(24939\) \(\beta_{15}\mathstrut +\mathstrut \) \(31494\) \(\beta_{14}\mathstrut +\mathstrut \) \(33856\) \(\beta_{13}\mathstrut +\mathstrut \) \(32868\) \(\beta_{12}\mathstrut +\mathstrut \) \(12804\) \(\beta_{11}\mathstrut +\mathstrut \) \(13298\) \(\beta_{10}\mathstrut -\mathstrut \) \(1092\) \(\beta_{9}\mathstrut +\mathstrut \) \(5271\) \(\beta_{8}\mathstrut -\mathstrut \) \(4231\) \(\beta_{7}\mathstrut +\mathstrut \) \(631\) \(\beta_{6}\mathstrut -\mathstrut \) \(21848\) \(\beta_{5}\mathstrut +\mathstrut \) \(31652\) \(\beta_{4}\mathstrut +\mathstrut \) \(50232\) \(\beta_{3}\mathstrut -\mathstrut \) \(26569\) \(\beta_{2}\mathstrut +\mathstrut \) \(41544\) \(\beta_{1}\mathstrut -\mathstrut \) \(28730\)
\(\nu^{14}\)\(=\)\(34545\) \(\beta_{15}\mathstrut +\mathstrut \) \(92328\) \(\beta_{14}\mathstrut +\mathstrut \) \(42883\) \(\beta_{13}\mathstrut +\mathstrut \) \(35508\) \(\beta_{12}\mathstrut +\mathstrut \) \(13085\) \(\beta_{11}\mathstrut +\mathstrut \) \(38161\) \(\beta_{10}\mathstrut +\mathstrut \) \(18504\) \(\beta_{9}\mathstrut +\mathstrut \) \(62007\) \(\beta_{8}\mathstrut -\mathstrut \) \(32868\) \(\beta_{7}\mathstrut -\mathstrut \) \(653\) \(\beta_{6}\mathstrut -\mathstrut \) \(13716\) \(\beta_{5}\mathstrut +\mathstrut \) \(33079\) \(\beta_{4}\mathstrut +\mathstrut \) \(74013\) \(\beta_{3}\mathstrut +\mathstrut \) \(702\) \(\beta_{2}\mathstrut +\mathstrut \) \(3857\) \(\beta_{1}\mathstrut +\mathstrut \) \(44462\)
\(\nu^{15}\)\(=\)\(176958\) \(\beta_{15}\mathstrut +\mathstrut \) \(234588\) \(\beta_{14}\mathstrut +\mathstrut \) \(247371\) \(\beta_{13}\mathstrut +\mathstrut \) \(237044\) \(\beta_{12}\mathstrut +\mathstrut \) \(104286\) \(\beta_{11}\mathstrut +\mathstrut \) \(107926\) \(\beta_{10}\mathstrut -\mathstrut \) \(8127\) \(\beta_{9}\mathstrut +\mathstrut \) \(46678\) \(\beta_{8}\mathstrut -\mathstrut \) \(35508\) \(\beta_{7}\mathstrut +\mathstrut \) \(3571\) \(\beta_{6}\mathstrut -\mathstrut \) \(151318\) \(\beta_{5}\mathstrut +\mathstrut \) \(228008\) \(\beta_{4}\mathstrut +\mathstrut \) \(373091\) \(\beta_{3}\mathstrut -\mathstrut \) \(188317\) \(\beta_{2}\mathstrut +\mathstrut \) \(272149\) \(\beta_{1}\mathstrut -\mathstrut \) \(206072\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−2.57541
−2.24781
−1.72082
−1.48078
−1.45188
−0.532475
−0.191617
−0.170601
0.102239
0.625678
1.24549
1.33134
1.79993
2.18919
2.37299
2.70451
−2.57541 0 4.63271 3.61540 0 −1.00000 −6.78030 0 −9.31112
1.2 −2.24781 0 3.05263 0.411504 0 −1.00000 −2.36611 0 −0.924982
1.3 −1.72082 0 0.961212 4.11468 0 −1.00000 1.78756 0 −7.08062
1.4 −1.48078 0 0.192706 −1.52223 0 −1.00000 2.67620 0 2.25409
1.5 −1.45188 0 0.107956 −0.584615 0 −1.00000 2.74702 0 0.848791
1.6 −0.532475 0 −1.71647 0.118172 0 −1.00000 1.97893 0 −0.0629236
1.7 −0.191617 0 −1.96328 −3.92161 0 −1.00000 0.759431 0 0.751445
1.8 −0.170601 0 −1.97090 0.206353 0 −1.00000 0.677439 0 −0.0352041
1.9 0.102239 0 −1.98955 −0.280585 0 −1.00000 −0.407889 0 −0.0286868
1.10 0.625678 0 −1.60853 0.920707 0 −1.00000 −2.25778 0 0.576066
1.11 1.24549 0 −0.448744 2.43368 0 −1.00000 −3.04990 0 3.03114
1.12 1.33134 0 −0.227522 2.92411 0 −1.00000 −2.96560 0 3.89299
1.13 1.79993 0 1.23975 −3.74624 0 −1.00000 −1.36840 0 −6.74297
1.14 2.18919 0 2.79257 −0.868150 0 −1.00000 1.73509 0 −1.90055
1.15 2.37299 0 3.63108 3.84175 0 −1.00000 3.87053 0 9.11644
1.16 2.70451 0 5.31438 1.33706 0 −1.00000 8.96376 0 3.61609
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 1.16
Significant digits:
Format:

Inner twists

This newform does not admit any (nontrivial) inner twists.

Atkin-Lehner signs

\( p \) Sign
\(3\) \(-1\)
\(7\) \(1\)
\(127\) \(1\)

Hecke kernels

This newform can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{2}^{\mathrm{new}}(\Gamma_0(8001))\):

\(T_{2}^{16} - \cdots\)
\(T_{5}^{16} - \cdots\)